Here \(F(x, dx)\)  is an arbitrary smooth positive function that is first-order homogeneous in its second argument.
After intensive study of Finsler, Cartan, Chern, and many others, this subject turned into a solid mathematical theory, see [1]}, [2]} and the references given therein. The classical Finsler's geometry is restricted to the positive defined case, i.e., to the Euclidean signature.
